Retraction Based Three-Dimensional Tracking of Object Movements

ABSTRACT

The technology disclosed relates to tracking movement of a real world object in three-dimensional (3D) space. In particular, it relates to mapping, to image planes of a camera, projections of observation points on a curved volumetric model of the real world object. The projections are used to calculate a retraction of the observation points at different times during which the real world object has moved. The retraction is then used to determine translational and rotational movement of the real world object between the different times.

RELATED APPLICATION

The present application is a continuation of and claims priority under35 U.S.C. §120 to U.S. Nonprovisional patent application Ser. No.14/162,720 entitled “SYSTEMS AND METHODS FOR TRACKING OBJECT MOVEMENTSIN THREE-DIMENSIONAL SPACE”, filed Jan. 23, 2014 (Attorney Docket No.LEAP 1029-2/LPM-008US), which claims priority under 35 U.S.C. §119(e)from U.S. Provisional Patent Application No. 61/755,660 entitled“SYSTEMS AND METHODS FOR TRACKING OBJECT MOVEMENTS IN THREE-DIMENSIONALSPACE”, filed Jan. 23, 2013 (Attorney Docket No. LEAP 1029-1/LPM-008PR),the entire disclosures of which are incorporated herein by reference forall purposes.

FIELD OF THE TECHNOLOGY DISCLOSED

The technology disclosed relates, in general, to characterizing andtracking movements of an object, and in particular implementations, todetermining positions and orientations of the moving object.

BACKGROUND

Motion capture has numerous applications. For example, in filmmaking,digital models generated using motion capture can be used to drive themotion of computer-generated characters or objects. In sports, motioncapture can be used by coaches to study an athlete's movements and guidethe athlete toward improved body mechanics. In video games or virtualreality applications, motion capture can be used to allow a person tointeract with a virtual environment in a natural way, e.g., by waving toa character, pointing at an object, or performing an action such asswinging a golf club or baseball bat.

The term “motion capture” refers generally to processes that capturemovement of a subject in three-dimensional (3D) space and translate thatmovement into, for example, a digital model or other representation.Motion capture is typically used with complex subjects that havemultiple separately articulating members whose spatial relationshipschange as the subject moves. For instance, if the subject is a walkingperson, not only does the whole body move across space, but the positionof arms and legs relative to the person's core or trunk are constantlyshifting. Motion capture systems are typically interested in modelingthis articulation.

Most existing motion-capture systems rely on markers or sensors worn bythe subject while executing the motion and/or on the strategic placementof numerous cameras in the environment to capture images of the movingsubject from different angles. Such systems tend to be expensive toconstruct. In addition, markers or sensors worn by the subject can becumbersome and interfere with the subject's natural movement. Further,systems involving large numbers of cameras tend not to operate in realtime, due to the volume of data that needs to be analyzed andcorrelated. Such considerations of cost, complexity and convenience havelimited the deployment and use of motion-capture technology.

Consequently, there is a need for an economical approach that capturesthe motion of objects in real time without attaching sensors or markersthereto.

SUMMARY

The technology disclosed relates to tracking movement of a real worldobject in three-dimensional (3D) space. In particular, it relates tomapping, to image planes of a camera, projections of observation pointson a curved volumetric model of the real world object. The projectionsare used to calculate a retraction of the observation points atdifferent times during which the real world object has moved. Theretraction is then used to determine translational and rotationalmovement of the real world object between the different times.

Implementations of the technology disclosed relate to methods andsystems for capturing the motion of one or more objects in 3D spaceusing, for example, a sphere model. In various implementations, anobject is modeled as a sphere, or any other kind of closed, 3D curvedvolume, distributed so as to volumetrically approximate the contour ofthe object; computation of the position and orientation of the volumedetermines the shape and/or movement of the object. The approachesdescribed herein capture the shape and/or movement of an object in 3Dspace and in real time without excessive cost or computationalcomplexity. Although the ensuing discussion refers to sphere-basedimplementations for convenience of presentation, it should be understoodthat any type of closed, 3D curved volume such as ellipsoid,hyperboloid, etc. may be used instead, and all such volumes are withinthe scope of the technology disclosed.

In one implementation, the position and orientation of a sphereassociated with an object of interest is determined based on multipleobservation points projected from the sphere; the observation points maybe obtained from, for example, light transmitted, reflected, orscattered from the sphere or shadows cast by the sphere. The observationpoints are first centrally projected onto multiple image sensors; theimage sensors may be located on two parallel image planes offset fromeach other; the image planes include a first set of centrally projectedpoints. A short time later, after the sphere has moved/rotated, a newset of points are centrally projected from the observation points of thesphere onto the same image planes. The location of the sphere center andthe orientation of the sphere relative to the image planes can then bedetermined based on the two sets of centrally projected points and thegeometry of the image planes captured by at least one camera. The 3Dstructure of the object and/or the movement thereof relative to theimage sensors is reconstructed by determining the location andorientation of the sphere. In some implementations, the object ismodeled as a collection of multiple spheres or any other closed curve.The structure and/or the movement of the object can then bereconstructed by assembling a collection of the determined locations andorientations of the multiple spheres.

In another implementation, the position and orientation of the object isfixed. Light emitted, reflected or scattered from the fixed object canbe projected onto multiple image sensors that are embedded in a portabledevice (e.g., a smart phone). Utilizing the approaches as describedabove, a relative movement between the portable device and the fixedobject can then be determined.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings, like reference characters generally refer to the likeparts throughout the different views. Also, the drawings are notnecessarily to scale, with an emphasis instead generally being placedupon illustrating the principles of the technology disclosed. In thefollowing description, various implementations of the technologydisclosed are described with reference to the following drawings, inwhich:

FIG. 1 illustrates a system for capturing image data according to animplementation of the technology disclosed.

FIG. 2 is a simplified block diagram of a gesture-recognition systemimplementing an image analysis apparatus according to an implementationof the technology disclosed.

FIG. 3A shows one implementation of image sensors collecting lightpattern images of an object of interest.

FIG. 3B illustrates one implementation of modeling an object or portionsof an object as one or more volumetric spheres.

FIG. 3C is one implementation of image sensors being disposed on twoparallel printed circuit board (PCB) bases.

FIG. 3D illustrates one implementation of monitoring light patterns fromobservation points on an object using cameras with planar image sensors.

FIG. 3E shows one implementation of monitoring light patterns fromobservation points on an object using a single camera with planar imagesensors.

FIG. 3F illustrates one implementation of partitioning space withindetection range of image sensors.

FIG. 3G is one implementation of grouping pixels in image sensors intomultiple pixel regions that correspond to multiple spatial partitions.

FIG. 4A depicts one implementation of capturing, in image planes of acamera, projections of observation points on a sphere model at times t0and t1.

FIG. 4B illustrates one implementation of calculating a refraction offour perimeter observation points at time t1 to their positions at t0.

FIG. 4C shows one implementation of determining a translation of asphere model between times t0 and t1.

FIG. 4D is one implementation of determining positions of centers of asphere model at the times t0 and t1.

FIG. 4E illustrates one implementation of a configuration of a spheremodel, image planes, and vanishing planes.

FIG. 5A shows one implementation of determining an intersection linebetween a first and second image plane of a camera.

FIG. 5B is one implementation of an edge-on view of a first and secondimage plane of a camera along with corresponding horizon lines.

FIG. 5C illustrates one implementation of calculating an orthogonaldistance between horizon lines in a first and second image plane of acamera and an intersection line between the first and second imageplanes.

FIG. 6 illustrates one implementation of determining a range of valuesof an angle between a first and second image plane of a camera.

FIG. 7A depicts a representative method of identifying a shape andmovement of an object in 3D space in accordance with one implementationof the technology disclosed.

FIG. 7B illustrates a representative method of identifying the shapesand movements of multiple objects in 3D space in accordance with oneimplementation of the current technology disclosed.

FIG. 8 is a flowchart showing a method of tracking movement and rotationof a sphere in accordance with implementations of the technologydisclosed.

DESCRIPTION

Referring first to FIG. 1, which illustrates a system 100 for capturingimage data according to an implementation of the technology disclosed.System 100 includes a pair of image sensors 102, 104 coupled to animage-analysis system 106. In one implementation, the image sensors arepreferably deployed on a printed circuit board (PCB). In anotherimplementation, the image sensors are employed in cameras; the camerascan be any type of camera, including cameras sensitive across thevisible spectrum or, more typically, with enhanced sensitivity to aconfined wavelength band (e.g., the infrared (IR) or ultraviolet bands);more generally, the term “camera” herein refers to any device (orcombination of devices) capable of capturing an image of an object, animage of a shadow cast by the object, an image of specular light spotsreflected by the object, or an image of laser speckle projected on theobject and representing that image in the form of digital data. Whileillustrated using an example two camera implementation, otherimplementations are readily achievable using different numbers ofcameras or non-camera light sensitive image sensors or combinationsthereof. For example, line sensors or line cameras rather thanconventional devices that capture a two-dimensional (2D) image can beemployed. The term “light” is used generally to connote anyelectromagnetic radiation, which may or may not be within the visiblespectrum, and may be broadband (e.g., white light) or narrowband (e.g.,a single wavelength or narrow band of wavelengths).

The cameras are preferably capable of capturing video images (i.e.,successive image frames at a constant rate of at least 15 frames persecond), although no particular frame rate is required. The capabilitiesof the cameras are not critical to the technology disclosed, and thecameras can vary as to frame rate, image resolution (e.g., pixels perimage), color or intensity resolution (e.g., number of bits of intensitydata per pixel), focal length of lenses, depth of field, etc. Inimplementations, any cameras capable of focusing on objects within aspatial volume of interest can be used. For instance, to capture motionof the hand of an otherwise stationary person, the volume of interestmight be defined as a cube approximately one meter on a side.

In some implementations, the illustrated system 100 includes a source108, which can be disposed near or on an object 110, and controlled bythe image-analysis system 106. In one implementation, the source 108 isa light source. For example, the light sources can be infrared lightsources like infrared light-emitting diodes (LEDs) or lasers. The imagesensors 102, 104 can be sensitive to infrared light or a characteristiclight spectrum of the laser, respectively. Use of infrared light canallow the gesture-recognition system 100 to operate under a broad rangeof lighting conditions and can avoid various inconveniences ordistractions that may be associated with directing visible light intothe region where the person is moving. However, a particular wavelengthor region of the electromagnetic spectrum is required. In addition,useful arrangements can include narrow- and wide-angle illuminators fordifferent wavelength ranges. In one implementation, filters 120, 122 areplaced in front of the image sensors 102, 104 to filter out visiblelight so that only infrared light is registered in the images capturedby the image sensors 102, 104. In another implementation, the source 108is a laser having low intensity, below a threshold capable of causingharm to a human. For example, the low-intensity lasers may have a lowpower (e.g., less than 1 mW) and/or a large area (e.g., larger than 10mm²) beam. In some implementations, additional optics (e.g., a lens or adiffuser) may be employed to widen the laser beam (and further weakenthe laser intensity). The low-intensity laser beam is emitted towardsthe object 110; images of laser beams projected onto the object are thencaptured by the image sensors 102, 104.

In operation, the light source 108 is arranged to emit light towards aregion of interest 112 in which an object of interest (e.g., a part of ahuman body such as a hand 110) that can sometimes also include a tool orother objects of interest can be present; the image sensors 102, 104 areoriented toward the region 112 to capture a light pattern of the object110 or a portion of the specular beam projected onto the object 110. Theimage sensors 102, 104 can detect light transmitted, reflected, orscattered from the object 110. In some implementations, the operation oflight source 108 and/or the image sensors 102, 104 is controlled by theimage-analysis system 106, which can be, e.g., a computer system. Basedon the captured images, image-analysis system 106 determines theposition and/or motion of object 110.

FIG. 2 is a simplified block diagram of a computer system 200, whichimplements image-analysis system 106 (also referred to as an imageanalyzer) according to an implementation of the technology disclosed.Image-analysis system 106 can include or consist of any device or devicecomponent that is capable of capturing and processing image data. Insome implementations, computer system 200 includes a processor 202, amemory 204, an image sensor interface 206, a display 208, speakers 209,a keyboard 210, and a mouse 211. Memory 204 can be used to storeinstructions to be executed by processor 202 as well as input and/oroutput data associated with execution of the instructions. Inparticular, memory 204 contains instructions, conceptually illustratedas a group of modules described in greater detail below, that controlthe operation of processor 202 and its interaction with the otherhardware components. An operating system directs the execution oflow-level, basic system functions such as memory allocation, filemanagement and operation of mass storage devices. The operating systemmay be or include a variety of operating systems such as MicrosoftWINDOWS operating system, the Unix operating system, the Linux operatingsystem, the Xenix operating system, the IBM AIX operating system, theHewlett Packard UX operating system, the Novell NETWARE operatingsystem, the Sun Microsystems SOLARIS operating system, the OS/2operating system, the BeOS operating system, the MACINTOSH operatingsystem, the APACHE operating system, an OPENSTEP operating system, iOS,Android or other mobile operating systems, or another operating systemof platform.

The computing environment may also include otherremovable/non-removable, volatile/nonvolatile computer storage media.For example, a hard disk drive may read or write to non-removable,nonvolatile magnetic media. A magnetic disk drive may read from orwrites to a removable, nonvolatile magnetic disk, and an optical diskdrive may read from or write to a removable, nonvolatile optical disksuch as a CD-ROM or other optical media. Other removable/non-removable,volatile/nonvolatile computer storage media that can be used in theexemplary operating environment include, but are not limited to,magnetic tape cassettes, flash memory cards, digital versatile disks,digital video tape, solid state RAM, solid state ROM, and the like. Thestorage media are typically connected to the system bus through aremovable or non-removable memory interface.

Processor 202 may be a general-purpose microprocessor, but depending onimplementation can alternatively be a microcontroller, peripheralintegrated circuit element, a CSIC (customer-specific integratedcircuit), an ASIC (application-specific integrated circuit), a logiccircuit, a digital signal processor, a programmable logic device such asan FPGA (field-programmable gate array), a PLD (programmable logicdevice), a PLA (programmable logic array), an RFID processor, smartchip, or any other device or arrangement of devices that is capable ofimplementing the actions of the processes of the technology disclosed.

The image sensor interface 206 can include hardware and/or software thatenables communication between computer system 200 and the image sensors102, 104 shown in FIG. 1, as well as associated light sources such asthe light source 108 of FIG. 1. Thus, for example, the image sensorinterface 206 can include one or more data ports 216, 218 to which theimage sensors can be connected, as well as hardware and/or softwaresignal processors to modify data signals received from the image sensors(e.g., to reduce noise or reformat data) prior to providing the signalsas inputs to a motion-capture (“mocap”) program 214 executing onprocessor 202. In some implementations, the image sensor interface 206can also transmit signals to the image sensors, e.g., to activate ordeactivate the image sensors, to control the image sensors settings(image quality, sensitivity, etc.), or the like. Such signals can betransmitted, e.g., in response to control signals from processor 202,which may in turn be generated in response to user input or otherdetected events.

The image sensor interface 206 can also include controllers 217, 219, towhich light sources (e.g., light source 108) can be connected. In someimplementations, controllers 217, 219 supply operating current to thelight sources, e.g., in response to instructions from processor 202executing mocap program 214. In other implementations, the light sourcescan draw operating current from an external power supply (not shown),and controllers 217, 219 can generate control signals for the lightsources, e.g., instructing the light sources to be turned on or off orchanging the intensities. In some implementations, a single controllercan be used to control multiple light sources.

Instructions defining mocap program 214 are stored in memory 204, andthese instructions, when executed, perform motion-capture analysis onimages supplied from the image sensors connected to the image sensorinterface 206. In one implementation, mocap program 214 includes variousprocessor-executed modules, such as an object-detection module 222 andan object-analysis module 224. Object-detection module 222 can analyzeimages (e.g., images captured via image sensor interface 206) to detectedges of an object therein and/or other information about the object'slocation. Object-analysis module 224 can analyze the object informationprovided by object detection module 222 to determine the 3D positionand/or motion of the object (e.g., a user's hand). Examples ofoperations that can be implemented in code modules of mocap program 214are described below. Memory 204 can also include other informationand/or code modules used by mocap program 214.

Display 208, speakers 209, keyboard 210, and mouse 211 can be used tofacilitate user interaction with computer system 200. These componentscan be modified as desired to provide any type of user interaction. Insome implementations, results of light pattern capture using the imagesensor interface 206 and mocap program 214 can be interpreted as userinput. For example, a user can perform hand gestures that create lightpattern; the light pattern are analyzed using mocap program 214, and theresults of this analysis can be interpreted as an instruction to someother program executing on processor 202 (e.g., a web browser, wordprocessor, or other application). Thus, by way of illustration, a usermight use upward or downward swiping gestures to “scroll” a webpagecurrently displayed on display 208, to use rotating gestures to increaseor decrease the volume of audio output from speakers 209, and so on.

It will be appreciated that computer system 200 is illustrative and thatvariations and modifications are possible. Computer systems can beimplemented in a variety of form factors, including server systems,desktop systems, laptop systems, tablets, smart phones or personaldigital assistants, and so on. A particular implementation may includeother functionality not described herein, e.g., wired and/or wirelessnetwork interfaces, media playing and/or recording capability, etc. Insome implementations, one or more image sensors may be built into thecomputer rather than being supplied as separate components. Further, animage analyzer can be implemented using only a subset of computer systemcomponents (e.g., as a processor executing program code, an ASIC, or afixed-function digital signal processor, with suitable I/O interfaces toreceive image data and output analysis results).

While computer system 200 is described herein with reference toparticular blocks, it is to be understood that the blocks are definedfor convenience of description and are not intended to imply aparticular physical arrangement of component parts. Further, the blocksneed not correspond to physically distinct components. To the extentthat physically distinct components are used, connections betweencomponents (e.g., for data communication) can be wired and/or wirelessas desired.

Referring to FIG. 3A, in various implementations, image sensors 102, 104are operated to collect a sequence of light pattern images of the object302 or specular light spots or shadows 304 projected onto the object302. These images are then analyzed, e.g., using the image analysissystem 106, to determine the object's position and shape in 3D space. Inone implementation, the image sensors 102, 104 are deployed on a PCBbase 306; the PCB base 306 includes a raised platform 308 so that theimage sensors can detect light pattern images in a plane parallel to thebase 306 with an offset distance z. The image sensors 102, 104 detectlight patterns that are transmitted, reflected, or scattered frommultiple (e.g., five or fewer) observation points 310 on surface of theobject 302. Referring to FIG. 3B, the object 302 or parts of the objectmay be modeled as a sphere 312 or a collection of spheres 312, or otherclosed volumetric curve; by monitoring the translational and rotationalmotion of the sphere(s) 312, the position and orientation of the object302 can be determined as further described below.

FIG. 3C illustrates another implementation in which the image sensors102, 104 are disposed on two PCB bases that are parallel and separatedby a distance z. The configuration and/or number of PCB bases, however,is not limited to the implementations as described herein; the optimalconfiguration and/or number of PCB bases for a particular application isstraightforwardly determined so that the image sensors 102, 104 candetect light pattern images cast from the object 302.

In the implementation illustrated in FIG. 3D, the light patterns castfrom the observation points are monitored using cameras 314, 316, eachof which has a planar image sensor 102, 104. The planar image sensors102, 104 are parallel and offset from each other by a distance z.Cameras 314, 316 thus project light pattern images of the observationpoints 310 from the object surface onto the planar image sensors 102,104. In another implementation, a single camera is used to monitor theobservation points 310 on the object surface. In the implementationillustrated in FIG. 3E, a single camera 318 includes two image sensors102, 104 that are parallel but offset by a distance z; accordingly,light pattern images of the observation points 310 are projected ontotwo different planes defined by the image sensors 102, 104. The lightpattern images of the observation points 310 acquired using either thesingle camera 318 or the dual cameras 314, 316 are then analyzed todetermine the location and orientation of the object 302.

In various implementations, the object 302 is modeled as a single sphereor a collection of spheres 312; theoretically, an infinite number ofspheres can be used to construct the 3D model of the object 302. In oneimplementation, the 3D model includes spheres that are close-packed(i.e., each sphere is tangent to adjacent spheres). Because theclosed-packed spheres occupy the greatest fraction of space volume ofthe object 302 with a limited number of spheres, the shape and size ofthe object 302 can be accurately modeled with a fast processing time(e.g., milliseconds). If a higher detection resolution of the object 302is desired, the number of spheres used to model the object 302 may beincreased.

FIG. 3F shows how space within the detection range of the image sensors102, 104 may be divided into multiple partitions 324; a part of theobject 302 in each partition 324 can be reconstructed using a spherethat fits the size and location thereof. A collection of spheres in thepartitions 324 then determines the shape, size, and location of theobject 302. As shown in FIG. 3G, pixels of image sensors 102, 104 may begrouped to form multiple regions 326, 328, each of which corresponds toa spatial partition 324. For example, light transmitted from a part 330of the object 302 in the spatial partition 332 is projected onto theregion 334 of the image sensor 102 to activate the pixels therein.Positions of the activated pixels in the region 334 can identify thelocation and/or size of the object part 330 by modeling it as, forexample, a sphere 336. In one implementation, five pixels activated bythe light transmitted, reflected, or scattered from the object 306 inthe partition 332 are used to determine the location and/or size of thesphere 336. Movements of the activated pixels in the pixel region 334can determine the motion of the sphere 336 (or the object part 330)within the spatial partition 332.

Movements of the activated pixels may result from a moving object part,or from a shape/size change of the object 302. In some implementations,object movements are identified based on the average movement of theactivated pixels and a predetermined maximum threshold movement. If, forexample, the average movement of the five activated pixels is within thepredetermined maximum threshold, it can be inferred that the movementsof the activated pixels result from a motion of the object and,consequently, object motion can be determined based on the movements ofthe five activated pixels. If, however, the average movement of the fiveactivated pixels is larger than the predetermined maximum threshold, itcan be inferred that the shape or size of the object part has changedand a new sphere is constructed to reflect this change. In someimplementations, an angular rotation of the sphere is determined basedon movement of one of the five activated pixels (e.g., the fifthactivated pixel) in the image sensor. Again, if movement of the fifthactivated pixel exceeds the predetermined threshold, a new sphere shouldbe used to reconstruct the object.

A collection of spheres from the regions 326, 328 of the image sensors102, 104, respectively, thus provides a reconstructed object in 3Dspace. Because the size and shape of each space partition may beadjusted based on a desired detection resolution and/or image-sensingspeed, the size and shape of each region 326, 328 in the image sensors102, 104 may be adjusted accordingly.

Referring again to FIG. 3B, in some implementations, the entire object302 is modeled as a sphere 312; this simplifies tracking the movement ofthe object 302 in 3D space. In some implementations, there is more thanone object within the detection range of the image sensors 102, 104;each object is modeled as, for example, a sphere or any other closedcurve and its position, size and movement are independently trackedusing, for example, five activated pixels in a corresponding region ofthe image sensor.

The object-tracking approach as described above determines the positionand movement of the object 302 relative to the image sensors 102, 104.In various implementations, the position of the object 302 is fixed, andthe same tracking approach is used to determine the relative locationsand movements of the multiple sensors. For example, the object 302 maybe any device or object that can transmit, reflect or scatter light andhas a fixed position in 3D space; the image sensors or any type ofsensors that can detect any signals transmitted, reflected, or scatteredfrom the object 302 may be embedded in a portable device. By collecting,for example, signals (e.g., optical signals, acoustic signals, or anyother electromagnetic signals) from the object 302, the movement of theimage sensors (and thereby the portable device) relative to the object302 can be determined using the same tracking approach as describedabove. For purpose of illustration, the image sensors herein are assumedto have fixed positions and the tracking approach is utilized todetermine the movement of the object 302.

In various implementations, the object 302 is modeled as a sphere.Parameters that characterize the sphere (such as the location (x, y, z)of the sphere center in 3D space, the radius r and the rotational angleθ of the sphere 312) may be determined based on multiple observationpoints 310 on the sphere surface. In some implementations, fiveobservation points on the sphere surface are monitored using the imagesensors to determine the characteristic parameters of the sphere.Referring to FIG. 4A, at an initial time t=t0, a sphere 400A having acenter 402A located at a position X0=(x0, y0, z0) in 3D space and with arotational angle θ₀ is tracked by indirectly monitoring a collection of,for example, five observation points 404A, 406A, 408A, 410A, 412A on thespherical surface. The rotational angle may be defined as an identitymatrix of order 3, i.e.,

$\theta_{0} = {\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}.}$

In one implementation, four of the observation points 404A, 406A, 408A,410A are centrally projected onto a first image plane 414 (e.g., thexy-plane at z=0), and the fifth observation point 412A is centrallyprojected onto a second image plane 416 that is parallel but offset fromthe first image plane 414 by a distance z (e.g., the xy-plane at z=1).

For example, the points 418A, 420A, 422A, 424A which are centrallyprojected from the observation points 404A, 406A, 408A, 410A,respectively, onto the first image plane 414 can be defined byintersecting lines that connect the spherical center 402A with theobservation points 404A, 406A, 408A, 410A, respectively, with the imageplane 414; similarly, point 426A is centrally projected from the fifthobservation point 412A onto the image plane 416 by intersecting the linethat connects the sphere center 402A and the observation point 412A withthe image plane 416.

As used herein, a collection of the centrally projected points 418A,420A, 422A, 424A in the first image plane 414 is defined as a first setof projected points P0 at t=t0, and the centrally projected point 426Ain the second image plane 416 is defined as a projected point q0 att=t0. In a short time interval, Δt=t1−t0 (e.g., 10 ms), the sphere 400Amay move to a new location as indicated by sphere 400B, the movementoccurring over a distance D and/or through a rotational angle Δθ; thisresults in the spherical center 402B being located in a new positionX1=(x1, y1, z1) and having a new rotational angle, θ₁=θ₀+Δθ, and thefive observation points 404A, 406A, 408A, 410A, 412A displaced to newpositions 404B, 406B, 408B, 410B, 412B, respectively. Accordingly,points in the image planes 414, 416 that are centrally projected fromthe points 404B, 406B, 408B, 410B, 412B are located at new positions418B, 420B, 422B, 424B, 426B in the image planes 414, 416, respectively,due to the movement of the sphere. Again, a second set of centrallyprojected points P1 at t=t1 is defined by a collection of the points418A, 420A, 422A, 424A in the image plane 414, and the projected pointq1 in the image plane 416 at t=t0 is defined by the point 426A.

Referring to FIG. 4B, in some implementations, informationcharacterizing movement of the sphere in the short time interval Δt, aswell as parameters that characterize the sphere, are determined based onthe relationship between the two sets of centrally projected points P0and P1 obtained at t0 and t1, respectively. In one implementation, thesphere 400B is retracted to the position and orientation of the sphere400A at time t=t0. The radius r of the sphere is assumed to have aconstant value at t=t0 and t=t1=t0+Δt and the sphere center is locatedat the origin, i.e., (x0, y0, z0)=O=(0, 0, 0). Accordingly, an imageplane 428 upon which the second set of the centrally projected points P1(i.e., 418B, 420B, 422B, 424B) is located is tilted at an angle φ withrespect to the image plane 414; the image planes 414, 428 are denoted asΠ₀, Π₁, respectively, and intersect at a line L01 indicated at 430. Insome implementations, the retraction of the sphere can be achieved byrescaling the positions of the observation points based on the distancesbetween the sets of centrally projected points P0, P1 and the spherecenters X0, X1 at t=t0 and t=t1, respectively (i.e., ∥P₀−X₀∥ and∥P₁−X₁∥). The relationship between the two sets of centrally projectedpoints P0, P1 located in the image planes Π₀, Π₁, respectively, may thenbe determined based on two mapping matrices, ρ₁ ⁰ and ρ₀ ¹, as furtherdescribed below. Similarly, referring to FIG. 4C, upon retracting themovement of the sphere, an image plane 431 in which the centrallyprojected point 426B (q1) is located is tilted at an angle φ withrespect to the image plane 416 in which the centrally projected point426A (q0) is located.

The mapping matrix ρ₁ ⁰ maps the points in the image plane Π₀ to thepoints in the image plane Π₁ (i.e., ρ₁ ⁰: Π₀→Π₁). In variousimplementations, this is achieved by creating a line connecting a pointπ in the image plane Π₀ (i.e., πεΠ₀) with the origin, O(0,0,0), andintersecting the line with the image plane Π₁. In one implementation,the set of centrally projected points P0 in the plane Π₀ is selected tomap to the image plane Π₁; as a result, the mapped points of P0 onto theimage plane Π₁ are the set of projected points P1. The mapping matrixcan then be expressed as: ρ₁ ⁰: P₀

P₁. Using this mapping principle, the mapping matrix ρ₁ ⁰ can also maplines in the image plane Π₀ to lines in the image plane Π₁. The mappingbetween the sets of projected points P0, P1 or lines in the image planesΠ₀, Π₁ may be a linear transformation that satisfies two equations:

ρ₁ ⁰(ν₁+ν₂)=ρ₁ ⁰(ν₁)+ρ₁ ⁰(ν₂)  (1),

ρ₁ ⁰(αν)=αρ₁ ⁰(ν)  (2),

where ν₁ and ν₂ are any vectors in the image plane Π₀, and α is a scalarvalue. Therefore, the mapping matrix ρ₁ ⁰ can be constructed based onthe two sets of centrally projected points P0, P1. Similarly, theinverse mapping matrix ρ₀ ¹: Π₁→Π₀, which maps points in the image planeΠ₁ onto points in the image plane Π₀ by creating a line that connects apoint in the plane Π₁ to the origin, O(0,0,0), and intersecting the linewith the image plane Π₀, can be easily constructed based again on thecentrally projected points P0, P1. The parameters that characterize theposition, rotation, and movement of the sphere can then be extractedfrom the mapping matrices ρ₁ ⁰ and ρ₀ ¹ as further described below.

In some implementations, however, the mapping matrices ρ₁ ⁰ and ρ₀ ¹ areneither linear nor affine transformations that preserve lines andparallelism (i.e., mapping parallel lines to parallel lines) during themapping. In fact, there may be points in the image plane Π₀ that fail tomap onto the image plane Π₁. For example, referring to FIG. 4D, a linethat connects the origin O with a point π_(i) in the plane Π₀ isparallel to the plane Π₁, thereby failing to be mapped onto the imageplane Π₁.

In some implementations, any set of points in a plane H1 (indicated at432) that contains the origin O and is parallel to the plane Π₁ do notmap onto the image plane Π₁; the plane H1 may thus be defined as a“vanishing plane” for the central projection from the sphere to Π₁. Ifthe image planes Π₀ and Π₁ are not parallel, points on a horizon line(or line at infinity) h1 (indicated at 434) defined by the intersectionof the plane Π₀ with the vanishing plane H1 fail to be mapped to theimage plane Π₁. Similarly, points in a plane H0 (indicated at 436) thatcontains the origin O and is parallel to the image plane Π₀ cannot bemapped onto the image plane Π₁ since the planes H0 and Π₁ are parallel;the horizon line (or line at infinity) 438 defined by the intersectionbetween plane Π₁ and the vanishing plane H0 is denoted as h0. FIG. 4Edepicts a configuration of the sphere, image planes Π₀, Π₁, and thevanishing planes H0, H1 corresponding to Π₀, Π₁, respectively, in 3Dspace.

In various implementations, the projective transformations that preservethe incidence and cross-ratio of the mapping lines but not necessarilytheir parallelism are used to obtain the mapping matrices ρ₁ ⁰ and ρ₀ ¹of all points in the image planes Π₀ and Π₁, respectively. Any planehaving ordinary Cartesian coordinates in a standard 3D space can betransformed by an invertible 3×3 matrix using homogeneous coordinates ina projective space. In addition, parallel planes that do not intersect(or intersect at “infinity”) in the standard 3D space can be transformedto projective planes that intersect in the projective space using theprojective transformation. For example, parallel planes Π₁ and H1 can beconverted to projective planes Π ₁ and H ₁, respectively, thatintersect; accordingly, the projective transformation allows derivationsof the mapping matrices ρ₁ ⁰ and ρ₀ ¹.

In one implementation, the projective plane

(P²) is defined as a space containing all lines passing through theorigin O in Euclidean 3-space

³. A line {t·(x,y,z)|tε

} in the homogeneous coordinate system of the Euclidean space isassigned coordinates [x,y,z] if (x,y,z)≠(0,0,0). Additionally, the line[x,y,z] is defined as being equal to [u,v,w] if there is a nonzero realnumber t for which (x,y,z)=(tu,tv,tw). Accordingly, when z≠0, [x:y:z]can be written as

$\left\lbrack {\frac{x}{z}:{\frac{y}{z}:1}} \right\rbrack;$

similarly, when y≠0,

$\left\lbrack {x:{y:z}} \right\rbrack = \left\lbrack {\frac{x}{y}:{1:\frac{z}{y}}} \right\rbrack$

and when x≠0,

$\left\lbrack {x:{y:z}} \right\rbrack = {\left\lbrack {1:{\frac{y}{x}:\frac{z}{x}}} \right\rbrack.}$

In various implementations, the projective plane includes multiplesubset {[x:y:1]|x,yε

},\{[x:1:z]|x,zε

},\{[1:y:z]|y,zε

}; each subset represents a copy of the finite (or affine) plane and hasa complement line (i.e., {[x:y:0]|x,yε

},\{[x:0:z]|x,zε

},\{[0:y:z]|y,zε

}, respectively) that is a copy of the projective line

(P¹), also called “the line at infinity.”

When an affine mapping matrix is expressed as w=Av+b in the projectiveplane, where w, v, b are vectors in the plane and A is a two-dimensional(2D) linear mapping, the corresponding projective mapping can beconstructed by converting vector coordinates into homogeneouscoordinates. Particularly, for vectors, affine coordinates are appendedby the homogenizing coordinate “1” and the offset (or the inhomogeneousterm, e.g., b) is folded into the linear mapping, as shown in equations(3) and (4).

$\begin{matrix}{{\begin{bmatrix}w_{x} \\w_{y}\end{bmatrix} = {{\begin{bmatrix}A_{xx} & A_{xy} \\A_{yx} & A_{yy}\end{bmatrix}\begin{bmatrix}v_{x} \\v_{y}\end{bmatrix}} + \begin{bmatrix}b_{x} \\b_{y}\end{bmatrix}}},} & (3) \\{\begin{bmatrix}w_{x} \\w_{y} \\1\end{bmatrix} = {{\begin{bmatrix}A_{xx} & A_{xy} & b_{x} \\A_{yx} & A_{yy} & b_{y} \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}v_{x} \\v_{y} \\1\end{bmatrix}}.}} & (4)\end{matrix}$

Eq. (3) represents an inhomogeneous form of the mapping matrix whereasEq. (4) represents a homogeneous form of the same mapping matrix.

In various implementations, a projective transformation (e.g., 2D) isexpressed as a 3×3 matrix constructed based on the same principles asdescribe above. Because of the arbitrary scale factor in the definitionof the homogeneous coordinates, there are eight parameters, rather thanthe nine parameters, required to define the 2D linear projectivetransformation. In one implementation, information about these eightparameters is obtained from the two sets of points, P0, P1, centrallyprojected onto the image planes Π₀ and Π₁, respectively. For example,suppose the acquired projected points of P0 and P1 are given as:

$\begin{matrix}{\mspace{79mu} {{P_{0} = {\begin{bmatrix}x \\y\end{bmatrix}_{0} = \begin{bmatrix}0.24078 & 0.59101 & 0.80300 & 0.49654 \\0.48684 & 0.38450 & 0.66520 & 0.87281\end{bmatrix}}},}} & (5) \\{P_{1} = {\begin{bmatrix}x \\y\end{bmatrix}_{1} = {\begin{bmatrix}{- 0.87354} & {- 0.32131} & 0.28201 & {- 0.15563} \\{- 3.071292} & {- 4.07336} & {- 3.20744} & {- 2.0261}\end{bmatrix}.}}} & (6)\end{matrix}$

To determine the mapping matrices ρ₁ ⁰ and ρ₀ ¹, two stages ofcomputation may be used. In a first stage, two initial transformationsfP₀, fP₁ are computed by applying a canonical frame

$\left( {{i.e.},\left. \quad\begin{bmatrix}0 & 1 & 1 & 0 \\0 & 0 & 1 & 1\end{bmatrix} \right)} \right.$

to the centrally projected point sets P0 and P1; this can be expressedas:

fP ₀=Transform(P ₀)  (7),

fP ₁=Transform(P ₁)  (8).

In a second stage, ρ₁ ⁰ is determined by applying the forward initialtransformation of P1, i.e., fP₁, to the inverse initial transformationof P0, i.e., (fP₀)⁻¹; similarly, ρ₀ ¹ is determined by applying theforward initial transformation of P0, i.e., fP₀, to the inverse initialtransformation of P1, i.e., (fP₁)⁻¹. This can be given as:

ρ₁ ⁰ =fP ₁×(fP ₀)⁻¹  (9),

ρ₀ ¹ =fP ₀×(fP ₁)⁻¹  (10).

where the operator “×” refers to matrix multiplication, and the notation(A)⁻¹ refers to matrix inversion. As a result, ρ₁ ⁰ and ρ₀ ¹ can beexpressed as:

$\begin{matrix}{{\rho_{1}^{0} = \begin{bmatrix}1.84017 & 0.61053 & {- 1.61385} \\{- 0.92731} & 2.93033 & {- 4.27462} \\{- 0.17426} & 0.30899 & 0.89153\end{bmatrix}},} & (11) \\{\rho_{0}^{1} = {\begin{bmatrix}0.50197 & {- 0.13310} & 0.27047 \\0.20057 & 0.17348 & 1.19485 \\0.02860 & {- 0.08614} & 0.76042\end{bmatrix}.}} & (12)\end{matrix}$

Because the third homogeneous coordinate of a mapping matrix representsa potential denominator, the third rows of the projectivetransformations ρ₁ ⁰ and ρ₀ ¹ give horizon lines h1 and h0,respectively, that are mapped to infinity. In this example, the line

−0.17426x+0.30899y+0.89153=0  (13),

in the image plane Π₀ is mapped to infinity by the mapping matrix ρ₁ ⁰,and is the horizon line h₁; similarly, the line

0.02860x−0.08614y+0.76042=0  (14).

in the image plane Π₁ is mapped to infinity by the inverse mappingmatrix ρ₀ ¹, and is the horizon line h₀. Accordingly, data from the setsof centrally projected points P₀ and P₁ can determine the mappingmatrices ρ₁ ⁰, ρ₀ ¹ and horizon lines h₁, h₀.

Referring to FIG. 5A, in some implementations, additional informationsuch as the intersection line L01 502 between the image planes Π₀ 504and Π₁ 506 (i.e., L₀₁=Π₀∩Π₁) is required to determine the positions ofthe sphere center. In one implementation, the intersection line L01(indicated at 502) is determined based on the geometry of the imageplanes Π₀, Π₁ and their corresponding vanishing planes H0, H1. To solvefor L01, lines that contain orthogonal projection points 508, 510 of thesphere center 512 in the planes Π₀ and Π₁, respectively, are firstdetermined. However, there are infinite numbers of lines that passthrough the projection points 508, 510 in the image planes Π₀ and Π₁,respectively. To simplify the calculation, in one implementation, a lineH0 that passes through the projection point 508 and is parallel to thehorizon line h1 in the image plane Π₀ is selected; similarly, a line H1that passes through the projection point 510 and is parallel to thehorizon line h0 in the image plane Π₁ is chosen.

In addition, the projection lines 518, 520 that pass through the spherecenter 512 and are orthogonal to the lines H0 and H1, respectively, aredenoted as H0 ^(⊥), H1 ^(⊥). Referring to FIG. 5B which depicts anedge-on view of planes Π₀, Π₁, H0, and H1. In some implementations, anauxiliary line OQ is defined by a plane M passing through the spherecenter O and is parallel to the horizons lines h1 and h0. The plane Mintersects the image planes Π₀ and Π₁ in a line R and a line Q,respectively. Since the vanishing plane H0 is parallel to the imageplane Π₀, the triangles ΔOh1R and ΔOPQ are similar to each other, i.e.,ΔOh1R˜ΔOPQ, and the corresponding sides of the triangles areproportional. Because PQ= h₁L₀₁ , the distance between h1 and L01 alongthe direction of the plane H0 ^(⊥) is given as:

$\begin{matrix}{\overset{\_}{h_{1}L_{01}} = {\frac{\overset{\_}{OQ}}{\overset{\_}{OR}\;} \times {\overset{\_}{h_{1}R}.}}} & (15)\end{matrix}$

Referring to FIG. 5C, because the lines R and Q are parallel in theplane M, ΔOAB˜ΔOCD; accordingly, the distance ratio

$\frac{\overset{\_}{OQ}}{\overset{\_}{OR}\;}$

can be given as:

$\begin{matrix}{\frac{\overset{\_}{OQ}}{\overset{\_}{OR}\;} = {\frac{\overset{\_}{CD}}{AB}.}} & (16)\end{matrix}$

As a result, the orthogonal distance between the horizon line h1 and theintersection line L01 of the image planes Π₀ and Π₁ can be determinedbased on the following procedure: (i) choosing any point π_(h) that lieson the horizon line h1 in the image plane Π₀, (ii) defining twoparameters τ₁=π_(h)+H0 ^(⊥), and τ₂=τ₁+H0, and (iii) determining thedistance h₁L₀₁ based on the mapping matrix ρ₁ ⁰, τ₁, and τ₂ using thefollowing formula:

h ₁ L ₀₁ = ρ₁ ⁰(τ₁)ρ₁ ⁰(τ₂) ρ₁ ⁰(τ₁)ρ₁ ⁰(τ₂)  (17).

Similarly, the distance between the horizon line h0 and the intersectionline L01 is determined based on the following procedure: (i) choosingany point π_(h′) that lies on the horizon line h0 in the image plane Π₁,(ii) defining two parameters τ₁=π_(h′)+H1 ^(⊥), and τ₂=τ₁+H1, and (iii)determining the distance h₀L₀₁ based on the inverse mapping matrix ρ₀ ¹,τ₁, and τ₂ using the following formula:

h ₀ L ₀₁ = ρ₀ ¹(τ₁)ρ₀ ¹(τ₂) ρ₀ ¹(τ₁)ρ₀ ¹(τ₂)  (17).

In various implementations, the positions of the sphere centers O aredetermined by two axes N0, N1 that intersect precisely at two points(i.e., each has exactly one intersection point inside the realprojective space) where each axis crosses the line L01; the axes N0, N1lie in the image planes Π₀ and Π₁, respectively. In one implementation,the axes are obtained by intersecting a plane

passing through the sphere center O with the image planes Π₀ and Π₁(i.e., Π₀ ∩

and Π₁ ∩

, respectively); the plane

is thus orthogonal to lines that are intersections between any two linesof h0, h1, L01 and H₀∩H₁ due to the geometrical symmetry. In anotherimplementation, the axes are determined based on the mapping matrices ρ₁⁰ and ρ₀ ¹. Because ρ₁ ⁰ is a rational mapping that is differentiable, afirst-order approximation to the mapping ρ₁ ⁰ may be obtained via usinga linear mapping (e.g., the matrix J₁ ⁰) thereof to transform tangentvectors at π in the image plane Π₀ to tangent vectors at ρ₁ ⁰(π) in theimage plane Π₁.

When the plane H0 is rotated about the axis Π₀∩H₁, the rotating planeH0ROT sweeps from H0 to H1, and the lines of intersection H_(0ROT)∩Π₀and H_(0ROT)∩Π₁ are parallel to each other. In particular, H_(0ROT)∩Π₀remains parallel to h1 and H_(0ROT)∩Π₁ remains parallel to h0. SinceH0ROT is made up entirely of lines passing through the origin O, ρ₁ ⁰maps the lines formed by the first set of the centrally projected pointsP0 to the lines formed by the second set of the centrally projectedpoints P1. Additionally, because of the geometry of the set of planes{Π₀, Π₁, H0, H1}, including that planes Π₀ and H0 are parallel, planesΠ₁ and H1 are parallel, and all the lines of intersection among the fourplanes are also parallel, the matrix Jacobian J₁ ⁰ maps H0 along thesame direction as H1 (i.e., J₁ ⁰ H0 is parallel to H1). Similarly, J₀ ¹maps H1 along the same direction as H0 (i.e., J₀ ¹ H1 is parallel toH0). The axis N0 in the image planes Π₀ and Π₁ can then be characterizedby J₁ ⁰ H0 ^(⊥), which is parallel to and H1 ^(⊥), and J₁ ⁰H1 ^(⊥),which is parallel to H0 ^(⊥), respectively. Further, for any given linethat is parallel to H0 in the image plane Π⁰, the component of J₁ ⁰ H0^(⊥) along the direction of the plane H1 is proportional to the distancefrom the axis; this can be expressed as:

For ξεAxis(Π₀),π=ξ+t·H ₀,  (18),

(H ₁)^(T)×(J ₁ ⁰(π)H ₀ ^(⊥))∝t  (19).

Accordingly, the axis N0 that includes a point ξ thereon can bedetermined based on the following approach: (i) constructing a pointπ₁=π+H0 for any point πεΠ₀ and offsetting the point π₁ by one unitlength, parallel to the horizon line h1, (ii) computing the Jacobianmatrices J₁ ⁰(π) and J₁ ⁰(π₁) at π and π₁, respectively, (iii) computingξ=π−t·H0 using t=a/b, where a=H1T×J₁ ⁰(π)×H0 ^(⊥) and b=H1T×(J₁ ⁰(π)−J₁⁰(π₁))×H0 ^(⊥), and (iv) determining the axis N0 based on the point ξand the geometrical characteristic that N0 is parallel to H1 ^(⊥). Theoperator “×” herein refers to matrix multiplication, and the notation(A)T refers to matrix transposition. Based on the same principles, theaxis N1 that is in the image plane Π₁ can be determined.

In various implementations, based on the calculated data L01, h0, and h1and an assumption that the original sphere center is at the origin, thearrangement of four planes, Π₀, Π₁, H0, H1, can be constructed and as aresult, the centers of the spheres can be determined based on theprocedure: (i) computing the intersection Z0 of the axis N0 with theline L01 and the intersection Z1 of the axis N1 with the line L01,wherein Z0εΠ₀, and Z1εΠ₁, (ii) determining parameters ρ₀ and ρ₁ usingρ₀= L₀₁h₁ and ρ₁= L₀₁h₀ (i.e., the distance between the line ofintersection L01 and the horizon lines h₁ ⊂Π₀ and h₀ ⊂Π₁, respectively),and (iii) assuming H0 ^(⊥) and H1 ^(⊥) point from Z0 and Z1 toward h1and h0, respectively, the positions of the sphere centers are thus givenby:

X ₀ =Z ₀+ρ₀ cos(φ)×H ₀ ^(⊥)+ρ₀ sin(φ)×Π₀ ^(⊥)  (20),

X ₁ =Z ₁+ρ₁ cos(φ)×H ₁ ^(⊥)+ρ₀ sin(φ)×Π₁ ^(⊥)  (21),

where φ is the angle between the two image planes Π₀ and Π₁ (i.e.,φ=∠(Π₀,Π₁)) and satisfies 0⁰≦φ≦180⁰. When the parameters Z₀, Z₁, H₀^(⊥), H₁ ^(⊥), Π₀ ^(⊥), Π₁ ^(⊥) are given in the coordinates from thexy-plane and Π₀ ^(⊥), ⊥₁ ^(⊥) are both [0,0,1]^(T), the positions of thesphere centers X₀, X₁ are in the ordinary Cartesian coordinates.

Once the positions of the sphere centers X0, X1 are determined, theobservation points Σ0, Σ1 on the sphere surface can be obtained bynormalizing the radii S0, S1, directed from the sphere centers to thecentrally projected points P0, P1 (i.e., S₀=P₀−X₀, S₁=P₁−X₁), to a unitlength. In addition, the orientation angles of the spheres can becomputed as a 3D orthogonal matrix with determinant 1 (i.e., a properrotation in 3D space). In one implementation, the original orientationof the sphere θ₀ is defined as an identity matrix of order 3; therelative rotational angle Δθ is then computed as follows: (i)decomposing the radii S0, S1 using QR-factorization: Q₀R₀=S₀,Q₁R₁=S₁,(ii) adjusting the signs: let R0;3, R1;3 be the third columns of R0, R3,respectively, and set ε_(i)=Sign (R_(0;3)[i]×R_(1;3)[i]) for i=1, 2, 3;the sign adjustment Δ is defined as:

${\Delta = \begin{bmatrix}ɛ_{1} & 0 & 0 \\0 & ɛ_{2} & 0 \\0 & 0 & ɛ_{3}\end{bmatrix}},$

and (iii) computing the relative rotation angle Δθ using Δθ=Q₁×Δ×Q₁^(T).

Because the QR factorization does not necessarily preserve theorientation of corresponding axes when comparing one factorization withanother, the sign adjustment (Δ) in the second action is necessary. Inaddition, this adjustment is sufficient because the third column of thematrix R inherits the orientations assigned to the previous columnsduring the factorization.

As described above, when determining the positions of the spherecenters, there is one free parameter, φ, representing the angle betweenthe two image planes Π₀ and Π₁. In some implementations, this freeparameter φ is determined based on simple geometry of the fifthcentrally projected points 426A, 426B in the image planes 416, 431,respectively, (as shown in FIGS. 4A and 4C) and the plane

passing through the sphere center O, orthogonal to the image planes Π₀and Π₁. Referring to FIG. 6, the lines Oq₀ and Oq₁ lie on the sameprojective trajectory, thereby having the same slope. Accordingly, whenthe intersection point between L01 and the plane Π₀ is set as an origin,the sphere center O, q0, and q1 are expressed as follows:

$\begin{matrix}{{O\text{:}\mspace{14mu} \left( {{{- b} + {a\; {\cos (\theta)}}},{a\; {\sin (\theta)}}} \right)},} & (22) \\{{q_{0}\text{:}\mspace{14mu} \left( {{- r},{- z}} \right)},} & (23) \\{q_{1}\text{:}\mspace{14mu} {\left( {{{s\; {\cos \left( {\theta + \pi} \right)}} + {z\; {\cos \left( {\theta + \frac{3\; \pi}{2}} \right)}}},{{s\; {\sin \left( {\theta + \pi} \right)}} + {z\; {\sin \left( {\theta + \frac{3\; \pi}{2}} \right)}}}} \right).}} & (24)\end{matrix}$

where b= L₀₁h₁ , a= L₀₁h₂ , r= L₀₁{circumflex over (q)}₀ , and s=L₀₁{circumflex over (q)}₁ ; {circumflex over (q)}₀ and {circumflex over(q)}₁ are defined as the nearest points to q₀ and q₁ in the image planesΠ₀ and Π₁, respectively.

Assuming C=cos(θ) and equating the slopes of Oq₀ and Oq₁ leads to aquadratic equation:

αC ² +βC+γ=0  (25),

where α=−z⁴−(2ab+(a+s)²+(b−r)²)z²−(rs−bs+ar)², β=−2(a+b)(s−r+a+b)z², andγ=(z²−bz−az+rs−bs+ar)(z²+bz+az+rs−bs+ar). The angle φ can then becomputed by solving cos(θ) and using φ=π−θ. Because there may bemultiple solutions for the angle φ, in some implementations, it isnecessary to check the solution for consistency with the acquiredcentrally projected data P₀ and P₁.

FIG. 7A depicts a representative method 700 of identifying a shape andmovement of an object in 3D space in accordance with one implementationof the technology disclosed. In a first action 702, images of the objectare captured by at least one image sensor. In a second action 704,pixels in the image sensor are grouped into multiple regions. In eachregion, a plurality (typically at least five) activated pixels areselected to identify the location and size of a sphere that representsat least a part of the object in 3D space (in a third action 706).Spheres corresponding to different regions of the image sensor can thenbe assembled to reconstruct the entire object (in a fourth action 708).The movement of the object can then be determined based on motion of theindividual spheres (in a fifth action 710).

FIG. 7B depicts a representative method 720 of identifying the shapesand movements of multiple objects in 3D space in accordance with oneimplementation of the technology disclosed. In a first action 722,images of the objects are captured by at least one image sensor. In asecond action 724, pixels in the image sensor are grouped into multipleregions. Each region includes activated pixels for modeling one of theobjects in 3D space (in a third action 726) by identifying a locationand a size of a sphere. In a fourth action 728, the motion of eachsphere is determined based on movements of multiple activated pixels inthe image sensor regions. The movements of the multiple objects relativeto the image sensor can be tracked by tracking the motion of themultiple spheres in 3D space (in a fifth action 730).

A representative method 800 of tracking movement and rotation of asphere in accordance with implementations of the technology disclosed isshown in FIG. 8. In a first action 802, a set of observation points Σ0on the sphere surface is projected onto one or more parallel imageplanes, Π₀, Π₀′, at t=t0, forming a first set of projected points, P0,therein; the image planes may be, for example, the planes defined by theimage sensors. In a second action 804, a second set of observationpoints Σ1 are projected onto the same image planes at t=t1, forming asecond set of projected points P1. In a third action 806, theobservation points Σ1 are moved to overlap with the observation pointsΣ0; the second set of projected points P1 are located in one or moreparallel image planes, Π₁, Π₁′; the angle between the image planes Π₀and Π₁ is φ.

In a fourth action 808, two mapping matrices, ρ₁ ⁰ and ρ₀ ¹, aredetermined based on the sets of projected points, P0 and P1; inaddition, the horizon lines h0, h1, in the image planes, Π₀, Π₁,respectively, are determined based on the mapping matrices, ρ₁ ⁰ and ρ₀¹. In a fifth action 810, the horizon lines h0 and h1 determine theintersection line L01 between the two image planes Π₀ and Π₁. Based onthe information about the horizon lines h0, h1 and the intersection lineL01, the geometry of the image planes Π₀, Π₁ and the correspondingvanishing planes H0, H1 is determined (in a sixth action 812). In aseventh action 814, the angle φ between the image planes Π₀ and Π₁ isdetermined based on the fifth points in the image planes Π₀′ and Π₁′. Inan eighth action 816, two axes N0 and N1 that intersect precisely at twopoints where each axis crosses the line L01 are then determined based onthe mapping matrices, ρ₁ ⁰, ρ₀ ¹, and the geometry of the image planesΠ₀, Π₁ and vanishing planes H0, H1. The positions of the sphere centerscan then be determined based on the two axes N0, N1 for all solutions ofthe angle φ (in a ninth action 818).

In a tenth action 820, the relative rotational angle of the sphere fromt=t0 to t=t1 is determined based on the two sets of the projectedpoints, P0, P1, and the positions of the sphere centers for allsolutions of the angle φ. Finally, in an eleventh action 822, themapping error for each solution of φ is determined via comparing thereconstructed spheres with the acquired observation points Σ0 and Σ1.The solution having the minimum mapping errors is then selected and thesphere is reconstructed based thereon (in a twelfth action 824).

Particular Implementations

In one implementation, a method of tracking movement of an objectportion in three-dimensional (3D) space is described. The methodincludes selecting five observation points on a curved volumetric modelfitted to a portion of a real world object, wherein one of theobservation points is at a center of the curved volumetric model and theother four observation points are along a perimeter of the curvedvolumetric model, capturing at times t0 and t1 projections of at leastthe four perimeter observation points in at least a first image plane ofat least one camera, wherein the object moved between t0 and t1,calculating a retraction of the four perimeter observation points attime t1 to their positions at to, including using the four perimeterobservation points to map the captured projections of the perimeterobservation points at t0 to a first image plane, and calculateorientation of a second image plane such that the captured projectionsof the perimeter observation points at t1 to the second image plane arecollinear with respective projections in the first image plane withrespect to the retracted center observation point, and determining atleast translation of the curved volumetric model between the times t0and t1 using the calculated orientation of the second image plane.

This method and other implementations of the technology disclosed caninclude one or more of the following features and/or features describedin connection with additional methods disclosed. In the interest ofconciseness, the combinations of features disclosed in this applicationare not individually enumerated and are not repeated with each base setof features. The reader will understand how features identified in thissection can readily be combined with sets of base features identified asimplementations.

In one implementation, the method also includes selecting a sixthobservation point along a perimeter of the curved volumetric model,capturing at the times t0 and t1 projections of the sixth observationpoint in a third image plane of the camera, and using a difference inposition of the sixth observation point in the third image planecombined with the calculated orientation of the second image plane tocalculate rotation of the curved volumetric model.

In one implementation, the retraction is calculated by rescalingpositions of the perimeter observation points based on distances betweenthe captured projections of the perimeter observation points at thetimes t0 and t1 and corresponding centers of the curved volumetric modelat the times t0 and t1.

In another implementation, the retraction is calculated by usingprojective transformations of a mapping matrix and an inverse mappingmatrix, which includes applying a canonical frame to projections of theperimeter observation points at t0 mapped to a first image plane andprojections of the perimeter observation points at t1 mapped to a secondimage plane, and applying a forward initial transformation of theprojections of the perimeter observation points at t0 to inverse initialtransformation projections of the perimeter observation points at t1.

In one implementation, the mapping matrix is constructed by mappingcaptured projections of the perimeter observation points at t0 tocaptured projections of the perimeter observation points at t1, whichincludes extending a line connecting a point on the first image planewith a center of the curved volumetric model at time t0, andintersecting the extended line with the second image plane.

In another implementation, the inverse mapping matrix is constructed bymapping captured projections of the perimeter observation points at t1to captured projections of the perimeter observation points at t0, whichincludes extending a line connecting a point on the second image plane,and intersecting the extended line with the second image plane.

In one other implementation, the method includes determining horizonlines on the first and second image planes that are mapped to infinitybased on the projective transformations of the mapping matrix andinverse mapping matrix. The method also includes determining positionsof the centers of the curved volumetric model at the times t0 and t1 byidentifying an intersection line between the first and second imageplanes and corresponding vanishing planes not mapped to any image plane.In one implementation, the intersection line is identifies based ongeometry of the first and second image planes and correspondingvanishing planes by determining planes lines that include orthogonalprojections of a center of the curved volumetric model in the first andsecond image planes and are parallel to horizon lines in the first andsecond image planes, extending projection lines that pass through thecenter of the curved volumetric model and are orthogonal to the planelines, creating a pair of intersection lines where the third image planeintersects with the first and second image planes responsive toextension of an auxiliary line in a third image plane that passesthrough the center of the curved volumetric model and is parallel to thehorizon lines in the first and second image planes, and calculatingorthogonal distance between the horizon lines in the first and secondimage planes and the intersection line between the first and secondimage planes responsive to construction of congruent triangles by thepair of intersection lines and the horizon lines in the first and secondimage planes.

In some implementations, the method also includes determining a range ofvalues of an angle between the first and second image planes based onthe sixth observation point along the perimeter of the curved volumetricmodel. It further includes determining positions of the centers of thecurved volumetric model at the times t0 and t1 by constructing two axesthat intersect at two points and also intersect the intersection linebased on the mapping matrix and inverse mapping matrix, geometry of thefirst and second image planes and corresponding vanishing planes, and arange of values of the angle between the first and second image planes.

In one implementation, the method includes determining a relativerotational angle of the curved volumetric from t0 to t1 based on thecaptured projections of the perimeter observation points at the times t0and t1 and positions of the centers of the curved volumetric model forthe range of values of the angle between the first and second imageplanes. It also includes calculating a mapping error for the range ofvalues of the angle by comparing new curved volumetric modelsconstructed based on the determined positions of the centers of thecurved volumetric model with the observation points along the perimeterof the curved volumetric model, and selecting a particular new curvedvolumetric model with minimum mapping error. It further includingcalculating positions of the observation points by normalizing, to aunit length, radii directed from corresponding centers of the new curvedvolumetric models to projections of the observations points.

Other implementations may include a non-transitory computer readablestorage medium storing instructions executable by a processor to performany of the methods described above. Yet another implementation mayinclude a system including memory and one or more processors operable toexecute instructions, stored in the memory, to perform any of themethods described above.

In another implementation, a method of tracking movement of an objectportion in three-dimensional (3D) space is described. The methodincludes capturing at times t0 and t1 projections of at least fourperimeter observation points on a curved volumetric model fitted to aportion of a real world object, calculating a retraction of the fourperimeter observation points at time t1 to their positions at t0 byusing projective transformations of a mapping matrix and an inversemapping matrix based on a first image plane to which projections of theperimeter observation points at t0 are mapped and a second image planeto which projections of the perimeter observation points at t1 aremapped, determining horizon lines on the first and second image planesthat are mapped to infinity based on the projective transformations ofthe mapping matrix and inverse mapping matrix, determining positions ofcenters of the curved volumetric model at the times t0 and t1 byidentifying an intersection line between the first and second imageplanes and corresponding vanishing planes not mapped to any image plane,determining a range of values of an angle between the first and secondimage planes based on a sixth observation point along the perimeter ofthe curved volumetric model, determining positions of the centers of thecurved volumetric model at the times t0 and t1 by constructing two axesthat intersect at two points and also intersect the intersection linebased on the mapping matrix and inverse mapping matrix, geometry of thefirst and second image planes and corresponding vanishing planes, andthe range of values of the angle between the first and second imageplanes, and determining a relative rotational angle of the curvedvolumetric from t0 to t1 based on the captured projections of theperimeter observation points at the times t0 and t1 and positions of thecenters of the curved volumetric model for the range of values of theangle between the first and second image planes.

This method and other implementations of the technology disclosed caninclude one or more of the following features and/or features describedin connection with additional methods disclosed.

In one implementation, the method also includes calculating a mappingerror for the range of values of the angle by comparing new curvedvolumetric models constructed based on the determined positions of thecenters of the curved volumetric model with the observation points alongthe perimeter of the curved volumetric model, and selecting a particularnew curved volumetric model with minimum mapping error. It furtherincludes calculating positions of the observation points by normalizing,to a unit length, radii directed from the corresponding centers of thenew curved volumetric models to projections of the observations points.

Other implementations may include a non-transitory computer readablestorage medium storing instructions executable by a processor to performany of the methods described above. Yet another implementation mayinclude a system including memory and one or more processors operable toexecute instructions, stored in the memory, to perform any of themethods described above.

In yet another implementation, a method of tracking movement of aportion of a real world object in three-dimensional (3D) space isdescribed. The method includes capturing images of a real world objectby at least one image sensor, wherein the real world object is in athree-dimensional (3D) space within detection range of the image sensor,grouping pixels in the image sensor into one or more pixel regions thatcorrespond to one or more spatial partitions of the 3D space,representing different portions of the real world object in the spatialpartitions by a collection of curved volumetric models, activating a setof pixels in a particular pixel region in response to light transmittedfrom a particular curved volumetric model in a corresponding spatialpartition, and determining a movement of a portion of the real worldobject represented by the curved volumetric model responsive tomovements of the activated pixels.

This method and other implementations of the technology disclosed caninclude one or more of the following features and/or features describedin connection with additional methods disclosed.

In one implementation, the method includes determining a location of thereal world object based on movements of the activated pixels. In anotherimplementation, the method includes determining a size of the real worldobject based on movements of the activated pixels.

In some implementations, if an average movement of the activated pixelsis within a threshold value, the method includes determining thatmovement of the activated pixels is in response to movement of the realworld object. In other implementations, if an average movement of theactivated pixels is above a threshold value, the method includesdetermining that movement of the activated pixels is in response to achange in shape or size of the real world object.

Other implementations may include a non-transitory computer readablestorage medium storing instructions executable by a processor to performany of the methods described above. Yet another implementation mayinclude a system including memory and one or more processors operable toexecute instructions, stored in the memory, to perform any of themethods described above.

In one implementation, a method of tracking a movement of an object in3D space is described. The method includes capturing first and secondsets of light pattern images generated by light cast from the object,each set of light pattern images comprising a plurality of projectionpoints, analyzing a projection point from each of the first and secondsets to computationally determine a set of parameters associating thesecond set of light pattern images with the first set of light patternimages, computationally determining a set of positions and relativemovements of the object based on the parameters and at least some of theprojection points in the first and second sets of light pattern images,each position and movement forming a pair associated with one of theparameters, determining a set of computing errors by comparing thedetermined positions and movements of the object with the first andsecond sets of light pattern images, each computing error beingassociated with one position and movement pair, and identifying aposition and a relative movement of the object in 3D space based atleast in part on the set of the computing errors.

This method and other implementations of the technology disclosed caninclude one or more of the following features and/or features describedin connection with additional methods disclosed.

In one implementation, the set of positions and relative movements ofthe object is determined based on the parameters and four of theprojection points in the first and second sets of light pattern images.In another implementation, the method includes determining a minimumcomputing error and identifying the position and movement of the objectbased thereon. In yet some implementations, the plurality of theprojection points are located in two parallel image planes offset fromone another. In one implementation, determining the set of positions andmovements of the object includes determining at least one mapping matrixfor mapping points from the second set of light pattern images to thefirst set of light pattern images.

In one implementation, the mapping matrix is determined by applying acanonical frame to the plurality of projection points in the first andsecond sets of light pattern images, and applying a forward initialtransformation of the plurality of projection points in the second setof light pattern images to an inverse initial transformation of theplurality of projection points in the first set of light pattern images.In another implementation, determining the set of positions andmovements of the object includes determining two horizon linesassociated with the first and second sets of light pattern images.

In another implementation, determining the set of positions andmovements of the object includes applying the mapping matrix to theplurality of projection points in the second set of light patternimages, and determining an intersection line between the first andsecond sets of light pattern images. In yet another implementation,determining the intersection lines includes computationally selecting apoint on one of the horizon lines, computationally defining twoparameters based on the selected point and a geometry of two imageplanes defined by the plurality of projection points in the first andsecond sets of light pattern images, computing a distance between theintersection line and the one of the horizon lines based on the twoparameters and the mapping matrix, and computationally determining theintersection line based on the computed distance and the one of thehorizon lines.

The method also includes computationally defining two axes that eachcross the intersection line, one of the axes lying in a first planedefined by the first set of light pattern images and the other axislying in a second plane defined by the second set of light patternimages and the mapping matrix, according to some implementations. Itfurther includes computationally determining one of the axes by andcomputationally determining the axis based on the linear mapping matrixand a geometry of two image planes defined by the plurality of theprojection points in the first and second sets of light pattern images.

In one implementation, the linear mapping matrix is a Jacobian matrix.In another implementation, the set of parameters is a set of rotationalangles between the first and second sets of light pattern images. In yetanother implementation, the movement includes a translational movementand a rotational movement.

Other implementations may include a non-transitory computer readablestorage medium storing instructions executable by a processor to performany of the methods described above. Yet another implementation mayinclude a system including memory and one or more processors operable toexecute instructions, stored in the memory, to perform any of themethods described above.

The terms and expressions employed herein are used as terms andexpressions of description and not of limitation, and there is nointention, in the use of such terms and expressions, of excluding anyequivalents of the features shown and described or portions thereof. Inaddition, having described certain implementations of the technologydisclosed, it will be apparent to those of ordinary skill in the artthat other implementations incorporating the concepts disclosed hereinmay be used without departing from the spirit and scope of thetechnology disclosed. Accordingly, the described implementations are tobe considered in all respects as only illustrative and not restrictive.

What is claimed is:
 1. A system of tracking movement of an object'sportion in a three-dimensional (3D) space, the system including: one ormore processors coupled to memory, the memory loaded with computerinstructions that, when executed on the processors, implement actionsincluding: selecting five observation points on a sphere model fitted toa portion of a real world object, wherein one of the observation pointsis at a center of the sphere model and the other four observation pointsare along a perimeter of the sphere model; capturing at times t0 and t1projections of at least the four perimeter observation points in atleast a first image plane of at least one camera, wherein the objectmoved between t0 and t1; calculating a retraction of the four perimeterobservation points at time t1 to their positions at to, including usingthe four perimeter observation points to: map the captured projectionsof the perimeter observation points at t0 to a first image plane; andcalculate orientation of a second image plane such that the capturedprojections of the perimeter observation points at t1 to the secondimage plane are collinear with respective projections in the first imageplane with respect to the retracted center observation point; anddetermining at least translation of the sphere model between the timest0 and t1 using the calculated orientation of the second image plane. 2.The system of claim 1, further configured to: select a sixth observationpoint along a perimeter of the sphere model; capture at the times t0 andt1 projections of the sixth observation point in a third image plane ofthe camera; and use a difference in position of the sixth observationpoint in the third image plane combined with the calculated orientationof the second image plane to calculate rotation of the sphere model. 3.The system of claim 1, further configured to calculate the retraction byrescaling positions of the perimeter observation points based ondistances between the captured projections of the perimeter observationpoints at the times t0 and t1 and corresponding centers of the spheremodel at the times t0 and t1.
 4. The system of claim 1, furtherconfigured to calculate the retraction using projective transformationsof a mapping matrix and an inverse mapping matrix, including: applying acanonical frame to projections of the perimeter observation points at t0mapped to a first image plane and projections of the perimeterobservation points at t1 mapped to a second image plane; and applying aforward initial transformation of the projections of the perimeterobservation points at t0 to an inverse initial transformationprojections of the perimeter observation points at t1.
 5. The system ofclaim 4, further configured to construct the mapping matrix by: mappingcaptured projections of the perimeter observation points at t0 tocaptured projections of the perimeter observation points at t1,including: extending a line connecting a point on the first image planewith a center of the sphere model at time t0; and intersecting theextended line with the second image plane.
 6. The system of claim 4,further configured to construct the inverse mapping matrix by: mappingcaptured projections of the perimeter observation points at t1 tocaptured projections of the perimeter observation points at t0;extending a line connecting a point on the second image plane; andintersecting the extended line with the second image plane.
 7. Thesystem of claim 4, further configured to determine horizon lines on thefirst and second image planes that are mapped to infinity based on theprojective transformations of the mapping matrix and inverse mappingmatrix.
 8. The system of claim 4, further configured to determinepositions of the centers of the sphere model at the times t0 and t1 byidentifying an intersection line between the first and second imageplanes and corresponding vanishing planes not mapped to any image plane.9. The system of claim 8, further configured to identify theintersection line based on geometry of the first and second image planesand corresponding vanishing planes by: determining planes lines thatinclude orthogonal projections of a center of the sphere model in thefirst and second image planes and are parallel to horizon lines in thefirst and second image planes; extending projection lines that passthrough the center of the sphere model and are orthogonal to the planelines; responsive to extension of an auxiliary line in a third imageplane that passes through the center of the sphere model and is parallelto the horizon lines in the first and second image planes, creating apair of intersection lines where the third image plane intersects withthe first and second image planes; and responsive to construction ofcongruent triangles by the pair of intersection lines and the horizonlines in the first and second image planes, calculating orthogonaldistance between the horizon lines in the first and second image planesand the intersection line between the first and second image planes. 10.The system of claim 2, further configured to determine a range of valuesof an angle between the first and second image planes based on the sixthobservation point along the perimeter of the sphere model.
 11. Thesystem of claim 8, wherein determining positions of the centers of thesphere model at the times t0 and t1 further includes: constructing twoaxes that intersect at two points and also intersect the intersectionline based on the mapping matrix and inverse mapping matrix, geometry ofthe first and second image planes and corresponding vanishing planes,and a range of values of the angle between the first and second imageplanes.
 12. The system of claim 10, further configured to determine arelative rotational angle of the sphere from t0 to t1 based on thecaptured projections of the perimeter observation points at the times t0and t1 and positions of the sphere centers for the range of values ofthe angle between the first and second image planes.
 13. The system ofclaim 10, further configured to: calculate a mapping error for the rangeof values of the angle by comparing new sphere models constructed basedon the determined positions of the centers of the sphere model with theobservation points along the perimeter of the sphere model; and select aparticular new sphere model with minimum mapping error.
 14. The systemof claim 13, further configured to calculate positions of theobservation points by normalizing, to a unit length, radii directed fromcorresponding centers of the new sphere models to projections of theobservations points.
 15. A system of tracking movement of an object'sportion in a three-dimensional (3D) space, the system including: one ormore processors coupled to memory, the memory loaded with computerinstructions that, when executed on the processors, implement actionsincluding: capturing at times t0 and t1 projections of at least fourperimeter observation points on a sphere model fitted to a portion of areal world object; calculating a retraction of the four perimeterobservation points at time t1 to their positions at t0 by: usingprojective transformations of a mapping matrix and an inverse mappingmatrix based on a first image plane to which projections of theperimeter observation points at t0 are mapped and a second image planeto which projections of the perimeter observation points at t1 aremapped; determining horizon lines on the first and second image planesthat are mapped to infinity based on the projective transformations ofthe mapping matrix and inverse mapping matrix; determining positions ofcenters of the sphere model at the times t0 and t1 by identifying anintersection line between the first and second image planes andcorresponding vanishing planes not mapped to any image plane;determining a range of values of an angle between the first and secondimage planes based on a sixth observation point along the perimeter ofthe sphere model; determining positions of the centers of the spheremodel at the times t0 and t1 by constructing two axes that intersect attwo points and also intersect the intersection line based on the mappingmatrix and inverse mapping matrix, geometry of the first and secondimage planes and corresponding vanishing planes, and the range of valuesof the angle between the first and second image planes; and determininga relative rotational angle of the sphere from t0 to t1 based on thecaptured projections of the perimeter observation points at the times t0and t1 and positions of the sphere centers for the range of values ofthe angle between the first and second image planes.
 16. The system ofclaim 15, further configured to: calculate a mapping error for the rangeof values of the angle by comparing new sphere models constructed basedon the determined positions of the centers of the sphere model with theobservation points along the perimeter of the sphere model; and select aparticular new sphere model with minimum mapping error.
 17. The systemof claim 15, further configured to calculate positions of theobservation points by normalizing, to a unit length, radii directed fromthe corresponding centers of the new sphere models to projections of theobservations points.
 18. A non-transitory computer readable medium,storing a plurality of instructions for programming one or moreprocessors to track movement of an object's portion in athree-dimensional (3D) space, the instructions, when executed on theprocessors, implementing actions including: selecting five observationpoints on a sphere model fitted to a portion of a real world object,wherein one of the observation points is at a center of the sphere modeland the other four observation points are along a perimeter of thesphere model; capturing at times t0 and t1 projections of at least thefour perimeter observation points in at least a first image plane of atleast one camera, wherein the object moved between t0 and t1;calculating a retraction of the four perimeter observation points attime t1 to their positions at t0, including using the four perimeterobservation points to: map the captured projections of the perimeterobservation points at t0 to a first image plane; and calculateorientation of a second image plane such that the captured projectionsof the perimeter observation points at t1 to the second image plane arecollinear with respective projections in the first image plane withrespect to the retracted center observation point; and determining atleast translation of the sphere model between the times t0 and t1 usingthe calculated orientation of the second image plane.
 19. Thenon-transitory computer readable medium of claim 18, further configuredto: select a sixth observation point along a perimeter of the spheremodel; capture at the times t0 and t1 projections of the sixthobservation point in a third image plane of the camera; and use adifference in position of the sixth observation point in the third imageplane combined with the calculated orientation of the second image planeto calculate rotation of the sphere model.
 20. The non-transitorycomputer readable medium of claim 18, further configured to calculatethe refraction using projective transformations of a mapping matrix andan inverse mapping matrix, including: applying a canonical frame toprojections of the perimeter observation points at t0 mapped to a firstimage plane and projections of the perimeter observation points at t1mapped to a second image plane; and applying a forward initialtransformation of the projections of the perimeter observation points att0 to an inverse initial transformation projections of the perimeterobservation points at t1.